A058711 Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 4, 1, 1, 14, 11, 1, 1, 51, 106, 26, 1, 1, 202, 1232, 642, 57, 1, 1, 876, 22172, 28367, 3592, 120, 1, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1

Offset

1,5

Comments

From Petros Hadjicostas, Oct 09 2019: (Start)

The old references had some typos, some of which were corrected in the recent ones. Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51; T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.

This triangular array is the same as A058710 except that it has no row n = 0 and no column k = 0.

(End)

Links

Table of n, a(n) for n=1..36.

W. M. B. Dukes, Tables of matroids.

W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.

W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.

W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.

Index entries for sequences related to matroids

Formula

From Petros Hadjicostas, Oct 09 2019: (Start)

T(n,1) = 1 for n >= 1.

T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2.

T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.

T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]

(End)

Example

Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,    1;
  1,    4,      1;
  1,   14,     11,       1;
  1,   51,    106,      26,      1;
  1,  202,   1232,     642,     57,     1;
  1,  876,  22172,   28367,   3592,   120,   1;
  1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
  ...

Crossrefs

Same as A058710 (except for row n=0 and column k=0).

Row sums give A058712.

Columns include (truncated versions of) A000012 (k=1), A058692 (k=2), A058715 (k=3).

Sequence in context: A267318 A050154 A179454 * A202906 A177984 A157013

Adjacent sequences: A058708 A058709 A058710 * A058712 A058713 A058714

Keyword

nonn,nice,tabl,more

Author

N. J. A. Sloane, Dec 31 2000

Extensions

Several values corrected by Petros Hadjicostas, Oct 09 2019

Status

approved